he profit from the production and sale of specialty golf hats is given by the function ​P(x)equals20xminus6000 where x is the number of hats produced and sold.

he profit from the production and sale of specialty golf hats is given by the function ​P(x)equals20xminus6000 where x is the number of hats produced and sold.

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The profit function \( P(x) = 20x - 6000 \) represents the profit from the production and sale of specialty golf hats, where \( x \) is the number of hats produced and sold.

Key Components of the Function:

  1. Coefficient of \( x \): The term \( 20x \) indicates that for each hat sold, the profit increases by $20.
  2. Constant Term: The constant \( -6000 \) represents the fixed costs associated with production, which must be covered before any profit can be realized.

Finding the Break-Even Point:

To determine the break-even point, where profit \( P(x) = 0 \):

\[ 20x - 6000 = 0 \]

Solving for \( x \):

\[ 20x = 6000 \] \[ x = \frac{6000}{20} = 300 \]

This means that 300 hats must be produced and sold to cover all costs, and any sales beyond this point will contribute to profit.

Profit Calculation:

  • For different values of \( x \):
    • If \( x = 0 \) (no hats sold), \( P(0) = 20(0) - 6000 = -6000 \) (loss of $6000).
    • If \( x = 300 \) (break-even), \( P(300) = 20(300) - 6000 = 0 \) (no profit, no loss).
    • If \( x = 400 \) (selling 400 hats), \( P(400) = 20(400) - 6000 = 8000 \) (profit of $8000).

Summary:

  • The profit increases linearly with each additional hat sold.
  • The break-even point is at \( x = 300 \).
  • The profit function indicates that below 300 hats sold results in a loss, while selling more than 300 hats results in profit.
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